I am not sure if you realize what you're arguing. I don't want to put words in your mouth, but I want to probe you to consider the implications.
My take on your post is that it's a very (classical) conservative view: in many cases, liberty undermines order, and that a sort of benevolent guiding force can keep order, which benefits us all. This is, like, Edmund Burke 101. It's anti-(classical) liberal. And in fact one of the major issues political conservatives were concerned with was the restriction of speech that agitated or disrupted harmony, so they agree with you on this.
Without trying to reductio ad absurdum you, let me suggest what I think this would look like. One possibility is that you are calling not for any solution, but for just a sort of feel-good agreement, a sort of principle among people, that we should go along to get along. I think that's tempting. We can all recognize this status quo of cultural cleavages leading to rag-tag boycotts is not working very well. But I am going to conclude that would fail.
The reason I conclude it would fail is more or less because that's what lead us here to begin with. So if you think that factionalism is a problem, and judging from this thread I would assume you do, then clearly people aren't going to voluntarily agree to do this.
On the other side of the spectrum, you could imagine a very heavy-handed intervention wherein wide swaths of issues are declared off-limits in the name of comity. Now, many people would say "hey, agreeing not to argue about something is a status quo politics that favours some majority opinion" but I'm actually going to sidestep that argument and just imagine that we could do this. In this theoretical world, you'd likely still have factions -- some of the original factions were tribal! Doing business with the people you know, the people who are closer to you, the people who are like you, who speak your language, preferentially... that's the original factionalism. My father-in-law was an engineer, and he worked with a few mechanics, and when my wife and I bought a used car, he referred us to the mechanics he worked with. They owed my father-in-law a favour or two, so we got discounted mechanic work. In return, we recommended him to some friends. And on and on.
Moreover, in all but the smallest towns, you typically have a choice of multiple merchants for a given good; so, like, I can choose to go to Sal's Cupcakes or Sweet Treats Cupcakes. Probably I don't have enough business to always go to both. And it might even be the case that the quality of their products and their prices are going to be around the same, so more or less my preference is going to be some kind of aesthetic -- meaning broadly more about the store's location, the friendliness of the service, things that basically don't matter -- preference. How do I force myself as a customer to affirmatively do business with everyone?
I think probably we're not going to be able to build a society where I am affirmatively obligated to do business with someone I find immoral, unless you're talking about a degree of central planning that more or less de-duplicates or centrally owns market sectors. It's true that every cupcake company was OmniCake by government fiat, and if I was only zoned to purchase from the one on Oak Street, then I wouldn't have a choice. But otherwise, I do. We do have this option, but I think here we'd be talking about a degree of totalizing central planning that more or less eclipses the USSR, DPRK, etc.
But of course there aren't just both ends of the spectrum, there's an in between. What would a "first step" of depoliticizing basic commerce look like? To my eye, any restriction on a consumer's speech is going to be more liberty-infringing than a restriction on a firm's speech. To me, it's less important that Chevy be able to express opinions without fear of reprisal than that I do.
So you might start by saying that any firm (over a certain size? if you're making this rule apply to sole proprietorships then we're basically back to square one here?) needs to shut the fuck up, broadly construed. They should not, either by law or just by convention, weigh in on any political issues. Although probably they want to weigh in on regulatory issues? Okay, so maybe they can weigh in on issues that affect them, but not "other" issues. Okay, it's true, justice issues affect staff and by affecting staff affect the firm, which is why FB and Apple weigh in on immigration and gay marriage. Crap. Now we're back to the status quo where firms are responsive to employee and customer political demands, which is the thing we're worried about in terms of factionalism.
Okay, maybe we just want to ban firms from weighing in on anything. No donations, no lobbying, no congressional testimony, no political research, no public communications about politics, no ads mentioning politics, they must legally no comment if asked, etc etc. Can they still donate to non-political charities, like can a charity shop still give money to the diabetes association? I don't know, that seems borderline, it's a hop skip and a jump from this to religion and politics.
This ends up being pretty tricky. Maybe the best outcome is that we sort of incrementally iterate on the sorts of issues we think ought be admissible and to what degree. Maybe there are some views that are so immoral they are deal-breakers, and others that bother me but I'll put up with them, and others that don't bother me but might bother you. And then we all, in a public sphere, advocate for which issues fall under which category. And this can change over time, so salient issues today aren't salient tomorrow. Shit, how do we keep ending up back here?
A slightly more serious suggestion: a lot of what's occurring here is the nationalization of politics. As firms are national and global in scale, issues are national and global, and so a lot of the comity you get for free from local homophily (meaning the tendencies of alike people to associate with one another, and thus for the people in your geographic vicinity to be more like you than people at random) has been eroded by the size of firms and the size of issues and the way in which consumers feel more solidarity on lines other than geography. You can solve this by breaking up firms, putting up market barriers, and basically structuring society around the idea that success comes from mutual investment in local firms; I win when I invest in Main Street in my neighborhood. This makes the social factions closer to the primordial ones of regional tribe and further from the modern ones. Of course, this would still disadvantage people who lose the geographic lottery and get born in a place they don't fit in well with. And this, by the way, explicitly subordinates efficiency and local advantage and mutual gains from trade to the idea that an organic solidarity of local homophily breeds comity and order.
My experience is that most people calling for comity more or less wish others thought more like them (and thus were bothered by the things they were, and not bothered by the things they are.) I think that might be a little naive.